Hyperfine optical vector analysis of ultra- narrowband optical bandpass filters based on coherent two-tone sweeping and fixed low-frequency detection

Yaowen Zhang; Yujia Zhang; Fei Yuan; Zhiyao Zhang; Shangjian Zhang; Heping Li; Yong Liu

doi:10.1364/OE.444539

1. Introduction

Optical filters with high quality factors (e.g., Q>10⁶) are key components to finely manipulate the optical spectrum in diverse applications such as optical communications, microwave photonics and optical signal processing [1–4]. Therefore, characterizing the magnitude and phase responses of optical filters with a high resolution is indispensable in the process of device fabrication and system design. Conventionally, magnitude and phase response characterization of optical filters is realized by using optical vector network analyzers (OVNAs) based on phase-shift method [5,6] or optical interferometric technology [7,8]. These methods employ tunable laser sources to sweep the optical device under test (ODUT), where the device response is extracted out from the heterodyning signal or the interference pattern. Due to the rough wavelength tuning resolution and the poor wavelength stability of the tunable laser sources, a measurement resolution generally larger than 100 MHz can be achieved [9], which is not sufficient for characterizing ultra-narrowband optical filters.

In recent years, OVNAs based on microwave photonic technology have been proposed to conquer the measurement resolution constraint [9–17]. The kernel of these methods is using finely tuning electro-optic modulation sideband controlled by an electrical vector network analyzer (EVNA) to sweep the ODUT, where the device response is obtained by measuring the relative magnitude and phase of the heterodyning signal between the optical carrier and the modulation sideband after system calibration. Up till now, both single-sideband (SSB) and double-sideband (DSB) modulations have been employed to realize high-resolution OVNAs, and a record-breaking high measurement resolution of 334 Hz has been achieved based on DSB [13]. However, these methods encounter difficulties in measuring ultra-narrow optical bandpass filters (OBPFs) with high out-of-band rejection ratios. For an OBPF with a bandwidth below 100 MHz, it is difficult to place the optical carrier just right into the passband due to the poor wavelength tunability of the commercially available off-the-shelf tunable laser sources. If the optical carrier is located outside the passband, it will be greatly attenuated due to the high out-of-band rejection ratio. Therefore, the heterodyning signal between the optical carrier and the modulation sideband is so weak that the filter response information cannot be successfully extracted out due to the limited dynamic range of the measurement setup. A promising method to acquire the frequency response of an ultra-narrowband OBPF is to separate the optical carrier and the modulation sideband into two paths, where the modulation sideband sweeps the ODUT, and the optical carrier is directly sent to the photodetector [16–17]. However, the phase response obtained by using this method is sensitive to external disturbance since two physical paths are employed to transmit the optical carrier and the modulation sideband separately. In principle, this problem can be solved by sweeping a two-tone optical probe signal with a fixed frequency interval across the passband together in a single physical path, where the frequency response of the ODUT can be obtained by detecting the heterodyne fixed low-frequency signal. OVNA based on coherent two-tone sweeping and fixed low-frequency detection has already been demonstrated [18,19]. Nevertheless, in [18], two single-tone optical signals with a fixed frequency interval propagate in two physical paths separately, which inevitably suffers from external disturbance. In [19], although the optical probe signal propagates through the ODUT in a single physical path, the two single-tone optical signals are generated by two independent fiber links. Hence, the external disturbance may also deteriorate the coherence of the generated two-tone optical probe signal.

In this paper, an OVNA based on coherent two-tone sweeping and fixed low-frequency detection is proposed and experimentally demonstrated to measure the frequency response of ultra-narrow OBPFs with high out-of-band rejection ratios. This scheme employs a dual-drive Mach-Zehnder modulator (DD-MZM) to generate two-tone optical probe signals with excellent coherence in a single fiber link. By using the generated two-tone optical signal (TTOS) with a fixed small-frequency interval to finely sweep the ODUT through electro-optic modulation in a single physical path, the magnitude and phase responses of the ODUT can be extracted out from the low-frequency heterodyning signal of the TTOS, which is with a high signal-to-noise ratio (SNR) in the frequency range of interest and free of external disturbance. In the experiment, the magnitude and group delay responses of a fiber-based Fabry-Perot tunable filter (FFP-TF) and a fiber-based Fabry-Perot interferometer (FFP-I) with 3-dB bandwidths of 1.5 GHz and 60 MHz, respectively, are successfully measured. In addition, the measurement uncertainty is theoretically and experimentally analyzed.

2. Operation principle

Figure 1 shows the schematic diagram of the proposed OVNA. A TTOS with a fixed small-frequency interval is generated by a home-built two-tone optical signal generator (TTOSG). The TTOSG is composed of a tunable laser source (TLS), an electro-optic DD-MZM and a tunable OBPF as shown in the dashed box of Fig. 1. In the TTOSG, two single-tone microwave signals with angular frequencies of ${\omega _{RF}}$ and ${\omega _{RF}} + \Delta \omega $ from a four-port electrical vector network analyzer (EVNA) are sent to the two radio-frequency (RF) ports of the DD-MZM, respectively. The DD-MZM is biased at its minimum transmission point to achieve carrier-suppressed double-sideband (DSB) modulation. After properly setting the passband of the tunable OBPF to select a group of 1^st-order modulation sidebands, a TTOS with a fixed angular frequency interval of $\Delta \omega $ is generated. The center wavelength of the TTOS can be roughly tuned by the TLS to reach the frequency measurement range, and finely swept by the EVNA to measure the magnitude and phase responses of the ODUT in the frequency range of interest. Then, the generated TTOS is amplified by using an erbium-doped fiber amplifier (EDFA) and split into two paths via an optical coupler. In the measurement path, the TTOS is firstly used to probe the ODUT, and then sent to a photodetector, i.e., PD1 in Fig. 1. The heterodyning signal at a fixed angular frequency of $\Delta \omega $ from PD1 carries the frequency response information of both the ODUT and the measurement system. In the reference path, the TTOS is directly sent to another photodetector, i.e., PD2 in Fig. 1. The heterodyning signal at a fixed angular frequency of $\Delta \omega $ from PD2 only carries the frequency response information of the measurement system. Finally, the two low-frequency heterodyning signals are measured by the EVNA, and the frequency response of the ODUT can be obtained after system calibration. In this scheme, the directly obtained phase information after system calibration is the phase variation within an angular frequency step of $\Delta \omega $. Therefore, the phase response of the ODUT should be reconstructed from this measured phase information. It should be noted that $\Delta \omega $ must be properly set to adapt to OBPFs with different bandwidths. In addition, it should be pointed out that the frequency response is obtained by fixed low-frequency detection, and the two optical probe tones propagate in a single physical path. Therefore, the measurement result is insensitive to external disturbance.

Fig. 1. Schematic diagram of the proposed OVNA. TTOSG: two-tone optical signal generator, TLS: tunable laser source, DD-MZM: dual-drive Mach-Zehnder modulator, EDFA: erbium-doped optical fiber amplifier, ODUT: optical device under test, PD: photodetector, EVNA: electrical vector network analyzer, TTOS: two-tone optical signal. Point A, B, C show the spectra of the signals after OBPF, ODUT and PD1, respectively.

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In the TTOSG, the optical field from the DD-MZM can be written as

(1)$$\begin{aligned} {E_1}(t )&= {e^{ - j({{\omega_c}t + \varphi (t )} )}}\left( {\sqrt {{P_1}} {e^{j{m_1}\sin ({{\omega_{RF}}t} )}} + {e^{j{\varphi_{dc}}}}\sqrt {{P_2}} {e^{j{m_2}\sin ({({{\omega_{RF}} + \Delta \omega } )t} )}}} \right)\\ &\textrm{ } = {e^{ - j({{\omega_c}t + \varphi (t )} )}}\sum\limits_{n ={-} \infty }^{ + \infty } {\left( {\sqrt {{P_1}} {J_n}({{m_1}} ){e^{jn{\omega_{RF}}t}} + {e^{j{\varphi_{dc}}}}\sqrt {{P_2}} {J_n}({{m_2}} ){e^{jn({{\omega_{RF}} + \Delta \omega } )t}}} \right)} \end{aligned}$$

where ${\omega _c}$ and $\varphi (t )$ are the angular frequency and the phase noise of the continuous-wave (CW) light from the TLS, respectively. ${P_1}$ and ${P_2}$ are the optical power split into the upper and lower branches of the DD-MZM, respectively. ${\varphi _{dc}}$ is the phase difference between the two arms in the DD-MZM, which is introduced by the direct-current (DC) bias voltage. ${m_1}$ and ${m_2}$ are the modulation indices of the two arms in the DD-MZM, respectively. ${J_n}(x )$ is the n^th-order Bessel function of the first kind. The DD-MZM is biased at its minimum transmission point to achieve carrier-suppressed DSB modulation, i.e., ${\varphi _{dc}} = \pi $. In addition, the tunable OBPF is properly set to select the -1^st-order modulation sidebands. Hence, the TTOS from the tunable OBPF can be written as

(2)$${E_2}(t )= {e^{ - j({{\omega_c}t + \varphi (t )} )}}({{A_1}{e^{j{\omega_{RF}}t}} + {B_1}{e^{j({{\omega_{RF}} + \Delta \omega } )t}}} )$$

where ${A_1} = \sqrt {{P_1}} {J_1}({{m_1}} )$ and ${B_1} ={-} \sqrt {{P_2}} {J_1}({{m_2}} )$.

The generated TTOS is then split into two paths, i.e., the measurement path and the reference path, as shown in Fig. 1. In the measurement path, the TTOS propagates through the ODUT, where the output optical field can be written as

(3)$${E_3}(t )= {e^{ - j({{\omega_c}t + \varphi ({t\textrm{ + }{\tau_{mea}}} )} )}}({H({{\omega_c} + {\omega_{RF}}} ){A_1}{e^{j{\omega_{RF}}({t + {\tau_{mea}}} )}} + H({{\omega_c} + {\omega_{RF}} + \Delta \omega } ){B_1}{e^{j({{\omega_{RF}} + \Delta \omega } )({t + {\tau_{mea}}} )}}} )$$

where $H(x )= {H_{sys}}(x ){H_{ODUT}}(x )$. ${H_{sys}}(x )$ and ${H_{ODUT}}(x )$ denote the complex frequency responses of the measurement system and the ODUT, respectively. ${\tau _{mea}}$ represents the time delay introduced by the external disturbance to the measurement path. It can be seen from Eq. (3) that the external disturbance introduces an identical time delay to the two optical tones in the TTOS. A low-speed PD, i.e. PD1 in Fig. 1, is used to detect ${E_3}(t )$. The complex amplitude of the photocurrent from PD1 at $\Delta \omega $ can be expressed as

(4)$${i_{mea}}({\Delta \omega } )= {R_1}({\Delta \omega } ){H^\ast }({{\omega_c} + {\omega_{RF}}} )H({{\omega_c} + {\omega_{RF}} + \Delta \omega } ){A_1}{B_1}{e^{j\Delta \omega {\tau _{mea}}}}$$

where ${R_1}(x )$ is the responsibility of PD1. Equation (4) indicates that the output signal from PD1 is with a fixed angular frequency of $\Delta \omega $ during frequency scanning, which carries the complex frequency responses of both the measurement system and the ODUT. In addition, the phase noise of the laser source $\varphi (t )$ has no influence on the output photocurrent, which means that the TTOS is coherent. Similarly, the complex amplitude of the photocurrent from PD2 at $\Delta \omega $ in the reference path can be expressed as

(5)$${i_{ref}}({\Delta \omega } )= {R_2}({\Delta \omega } )H_{sys}^\ast ({{\omega_c} + {\omega_{RF}}} ){H_{sys}}({{\omega_c} + {\omega_{RF}} + \Delta \omega } ){A_1}{B_2}{e^{j\Delta \omega {\tau _{ref}}}}$$

where ${R_2}(x )$ is the responsibility of PD2, and ${\tau _{ref}}$ represents the time delay introduced by the external disturbance to the reference path. Equation (5) indicates that the output signal from PD2 is also with a fixed angular frequency of $\Delta \omega $ during frequency scanning, which carries the complex frequency response of only the measurement system. In addition, it can be seen from Eq. (4) and Eq. (5) that the external disturbance introduces phase shifts of $\Delta \omega {\tau _{mea}}$ and $\Delta \omega {\tau _{ref}}$ to the generated photocurrents in the measurement path and the reference path, respectively. In the laboratory environment, the vibration and stress disturbance can be avoided. Hence, the main external disturbance is attributed to the temperature variation. The measurement path generally involves a maximum fiber length of 10 m. The temperature coefficient of the optical fiber is about 40 ps/km/℃. Therefore, for a temperature variation of 1℃ and a photocurrent with a frequency of 100 MHz, the maximum phase variation is calculated as

(6)$$10m \times 1^\circ C \times 40ps/km/^\circ C \times 100MHz \times 2\pi = 8\pi \times {10^{ - 5}}rad$$

which is tiny. Therefore, the external disturbance has a negligible influence on the measurement. The external disturbance-induced phase shifts of $\Delta \omega {\tau _{mea}}$ and $\Delta \omega {\tau _{ref}}$ can be ignored in Eq. (4) and Eq. (5), respectively.

A calibration procedure should be implemented to extract out the complex frequency response of the ODUT. Based on Eq. (4) and Eq. (5), the complex frequency response of the ODUT can be calculated as

(7)$$H_{ODUT}^\ast ({{\omega_c} + {\omega_{RF}}} ){H_{ODUT}}({{\omega_c} + {\omega_{RF}} + \Delta \omega } )= \frac{{{i_{mea}}({\Delta \omega } ){R_2}({\Delta \omega } )}}{{{i_{ref}}({\Delta \omega } ){R_1}({\Delta \omega } )}}$$

where ${{{R_2}({\Delta \omega } )} / {{R_1}({\Delta \omega } )}}$ is a constant for a fixed $\Delta \omega $. In the measurement, $\Delta \omega $ is set to be much smaller than the passband width of the ODUT. Hence, an important approximation can be made as follows

(8)$$\begin{aligned} &H_{ODUT}^\ast ({{\omega_c} + {\omega_{RF}}} ){H_{ODUT}}({{\omega_c} + {\omega_{RF}} + \Delta \omega } )\\ &= |{{H_{ODUT}}({{\omega_c} + {\omega_{RF}}} )} ||{{H_{ODUT}}({{\omega_c} + {\omega_{RF}} + \Delta \omega } )} |{e^{j[{{\phi_{ODUT}}({{\omega_c} + {\omega_{RF}} + \Delta \omega } )- {\phi_{ODUT}}({{\omega_c} + {\omega_{RF}}} )} ]}}\\ &\approx {\left|{{H_{ODUT}}\left( {{\omega_c} + {\omega_{RF}} + \frac{{\Delta \omega }}{2}} \right)} \right|^2}{e^{ - jG{D_{ODUT}}\left( {{\omega_c} + {\omega_{RF}} + \frac{{\Delta \omega }}{2}} \right)\Delta \omega }} \end{aligned}$$

where ${\phi _{ODUT}}(\omega )$ and $G{D_{ODUT}}(\omega )$ represent the phase response and group delay of the ODUT at the angular frequency $\omega $, respectively. Through substituting Eq. (8) into (7), the response of the ODUT at ${\omega _c} + {\omega _{RF}} + {{\Delta \omega } / 2}$ can be calculated as

(9)$${\left|{{H_{ODUT}}\left( {{\omega_c} + {\omega_{RF}} + \frac{{\Delta \omega }}{2}} \right)} \right|^2}{e^{ - jG{D_{ODUT}}\left( {{\omega_c} + {\omega_{RF}} + \frac{{\Delta \omega }}{2}} \right)\Delta \omega }} = \frac{{{i_{mea}}({\Delta \omega } ){R_2}({\Delta \omega } )}}{{{i_{ref}}({\Delta \omega } ){R_1}({\Delta \omega } )}}$$

Therefore, through finely sweeping the TTOS across the passband of the ODUT and measuring the two low-frequency heterodyning signals by using a four-port EVNA, the frequency response of the ODUT can be obtained based on Eq. (9).

It should be pointed out that the optical field from the tunable OBPF actually contains other frequency components except for the wanted -1^st-order modulation sidebands due to the unsatisfactory filtering performance of the commercially-available tunable OBPF. These additional frequency components include residual optical carrier, +1^st-order modulation sidebands and high-order modulation sidebands. Thereinto, residual optical carrier and high-order modulation sidebands have no influence on the measurement if the measurement angular frequency of the EVNA is set to be $\Delta \omega $ and its intermediate analysis angular bandwidth is set to be much smaller than $\Delta \omega $. In addition, residual +1^st-order modulation sidebands also have negligible influence on the measurement since they are actually located outside the passband of the ODUT, and are greatly attenuated.

3. Experimental results

A proof-of-concept experiment is carried out to verify the feasibility of the proposed scheme. A tunable laser source with a nominal linewidth of ∼500 kHz (Santec TSL-510) is employed to generate CW optical carrier with a power of 10 dBm. TTOS is generated by using a 40-Gbit/s DD-MZM (EOspace AZ-DD-0VPP-40) and a tunable OBPF (Santec OTF-350), where the DD-MZM is driven by two single-tone RF signals with a fixed small frequency interval from a four-port EVNA (Keysight N5225A). The generated TTOS is amplified by an EDFA (Amonics, AEDFA-PA-30) and then split into two paths via a 3-dB optical coupler. Two PDs (HP 11982A) are employed to achieve photodetection in the measurement and reference paths, and the photocurrents from the two PDs are measured by the EVNA.

Firstly, the performance of the generated TTOS is tested. Figure 2(a) shows the spectra of the optical signals before and after the tunable OBPF, where two single-tone RF signals with frequencies of 12 GHz and 12.1 GHz are applied to the DD-MZM, and the tunable OBPF is properly set to select the -1^st-order modulation sidebands. It can be seen from Fig. 2(a) that the suppression ratio between the ±1^st-order modulation sidebands after the tunable OBPF reaches ∼38 dB, which is favorable for reducing the measurement errors induced by the residual +1^st-order modulation sidebands. This suppression ratio will be much higher for RF signals with higher frequencies due to the rapid edge roll-off (i.e., 200 dB/nm) of the tunable OBPF as shown by the dashed line in Fig. 2(a). The employed optical spectrum analyzer (YOKOGAWA AQ6370C) is with a limited wavelength resolution of 0.02 nm, i.e., ∼2.5 GHz at 1550 nm. Hence, the two frequency components in each modulation sideband cannot be distinguished in Fig. 2(a). In order to observe the detail of the -1^st-order modulation sidebands, a frequency-shift heterodyne scheme incorporating a 100 MHz AOM is employed [20]. Figure 2(b) exhibits the heterodyne electrical spectrum corresponding to the -1^st- and -2^nd-order modulation sidebands. It can be seen from Fig. 2(b) that there is no 3^rd-order intermodulation components at 11.8 GHz and 12.1 GHz. Moreover, the -1^st- and -2^nd-order modulation sidebands will beat in the PD to produce photocurrents with different frequencies of 100 MHz and 200 MHz, respectively, as shown in Fig. 2(c). Through setting the analysis frequency and the analysis bandwidth of the EVNA to be 100 MHz and much smaller than 100 MHz, respectively, the influence of the heterodyne signals between the high-order modulation sidebands can be eliminated. Therefore, there is no measurement error resulting from intermodulation and modulation harmonics.

Fig. 2. (a) Spectra of the optical signals before and after the tunable OBPF, (b) heterodyne electrical spectrum corresponding to the -1^st- and -2^nd-order modulation sidebands based on frequency-shift heterodyne scheme incorporating a 100 MHz AOM, and (c) electrical spectrum of the photocurrent from the PD.

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An FFP-TF (LUNA FFP-TF) with a 3-dB passband width of about 1.5 GHz is used as the ODUT. The red lines in Fig. 3(a) and (b) show the magnitude and group delay responses measured by the proposed OVNA with a fixed frequency interval of 100 MHz. The measurement results indicate that the 3-dB passband width and the maximum group delay of the FFP-TF are 1.5 GHz and 0.22 ns, respectively. For comparison, the magnitude response is also measured by using other two methods. The first one is the optical spectrum analysis (OSA) method, which is achieved by using the optical spectrum analyzer and an amplified spontaneous emission source (Golight ASE-C + L). The blue line in Fig. 3(a) shows the measurement result, which indicates that the limited wavelength resolution of the employed optical spectrum analyzer, i.e., ∼2.5 GHz at 1550 nm, is not sufficient for characterizing this FFP-TF. The other method is based on SSB modulation scheme [17], where the optical carrier and the modulation sideband are split into two physical paths, and only the latter propagates through the ODUT. The gray line in Fig. 3(a) exhibits the measurement result, which fits in with that obtained by using the proposed method. The ripples in the measured magnitude response by using the SSB modulation method is attributed to the phase noise of the laser source and the different external disturbance to the two physical paths. In addition, the intrinsic group delay response of the FFP-TF is calculated from the measured magnitude response based on the well-known Kramers-Kronig (KK) relation since the FFP-TF is with a minimum phase-shift characteristic [21,22]. The blue line in Fig. 3(b) shows the calculated group delay response, which fits in with the measurement result by using the proposed OVNA. The gray line in Fig. 3(b) is the measured group delay response by using the SSB modulation method [17]. Apparently, the SSB modulation method cannot achieve group delay measurement due to the phase noise of the laser source and the environmental vibration. These results indicate that the proposed OVNA has the ability to measure both the magnitude and phase responses of a narrowband OBPF.

Fig. 3. Measured (a) magnitude and (b) group delay responses of an FFP-TF. The inset in Fig. 3(b) is the zoom-in view of the group delay response measured by the proposed method.

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Another experiment is carried out to verify the high-resolution measurement capability of the proposed OVNA. In the experiment, an FFP-I (LUNA FFP-I) with a nominal 3-dB passband width of less than 100 MHz is used as the ODUT. In order to finely measure the frequency responses of this FFP-I, the frequency interval of the TTOS should be smaller than 10 MHz. However, the minimum detectable frequency of the employed EVNA is 10 MHz, which is unable to directly measure the two low-frequency heterodyning signals from the PDs. Therefore, two mixers are inserted into the electrical backend to achieve up-conversion of the two low-frequency heterodyning signals from the PDs. Figure 4 presents the up-conversion architecture, where a single-tone microwave signal with an angular frequency of ${\omega _{up}}$ from a microwave source is used as the local signal. Based on the up-conversion architecture, the two low-frequency heterodyning signals from the PDs at $\Delta \omega $ is up-converted to ${\omega _{up}} + \Delta \omega $, where the measurement frequency of the EVNA is also set to be ${\omega _{up}} + \Delta \omega $. The red lines in Fig. 5(a) and (b) exhibit the measured magnitude and group delay responses of the FFP-I by using the proposed OVNA, respectively. The measurement results indicate that the 3-dB passband width and the maximum group delay of the FFP-I are 60 MHz and 6 ns, respectively. For comparison, the intrinsic group delay response of the FFP-I is calculated from the measured magnitude response based on KK relation, which is also shown in Fig. 5 (b). It can be seen from Fig. 5(b) that the measured group delay response and the retrieved one are consistent in the passband of the FFP-I. These experimental results indicate that the proposed OVNA can be used to measure the frequency response of an OBPF with an ultra-narrow passband width.

Fig. 4. Up-conversion architecture. MS: microwave source.

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Fig. 5. Measured (a) magnitude and (b) group delay responses of an FFP-I.

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4. Measurement uncertainty

The measurement uncertainty of the proposed OVNA mainly originates from the approximation in Eq. (8) and the measurement error of the EVNA.

4.1 Measurement uncertainty induced by the approximation in Eq. (8)

The magnitude response of the ODUT at ${\omega _0}$, ${\omega _0} + {{\Delta \omega } / 2}$ and ${\omega _0} - {{\Delta \omega } / 2}$ are denoted by $A({{\omega_0}} )$, $A({{\omega_0} + {{\Delta \omega } / 2}} )$ and $A({{\omega_0} - {{\Delta \omega } / 2}} )$, respectively. Based on Eq. (8), an approximation is made as

(10)$${A^2}({{\omega_0}} )\approx A\left( {{\omega_0} + \frac{{\Delta \omega }}{2}} \right)A\left( {{\omega_0} - \frac{{\Delta \omega }}{2}} \right)$$

In order to analyze the measurement uncertainty induced by this approximation, $A({{\omega_0} + {{\Delta \omega } / 2}} )$ and $A({{\omega_0} - {{\Delta \omega } / 2}} )$ are expanded by using Taylor expansion as

(11)$$A\left( {{\omega_0} + \frac{{\Delta \omega }}{2}} \right) = A({{\omega_0}} )+ A^{\prime}({{\omega_0}} )\frac{{\Delta \omega }}{2} + \frac{{A^{\prime\prime}({{\omega_0}} )}}{2}{\left( {\frac{{\Delta \omega }}{2}} \right)^2} + o\left[ {{{\left( {\frac{{\Delta \omega }}{2}} \right)}^2}} \right]$$

(12)$$A\left( {{\omega_0} - \frac{{\Delta \omega }}{2}} \right) = A({{\omega_0}} )- A^{\prime}({{\omega_0}} )\frac{{\Delta \omega }}{2} + \frac{{A^{\prime\prime}({{\omega_0}} )}}{2}{\left( {\frac{{\Delta \omega }}{2}} \right)^2} + o\left[ {{{\left( {\frac{{\Delta \omega }}{2}} \right)}^2}} \right]$$

where $A^{\prime}(x )$ and $A^{\prime\prime}(x )$ represent the 1^st- and 2^nd-order differentials of $A(x )$, respectively. $o(x )$ represents the high-order expansion error, which is infinitesimal. Based on Eq. (11) and Eq. (12), the following calculation can be made

(13)$$A\left( {{\omega_0} + \frac{{\Delta \omega }}{2}} \right)A\left( {{\omega_0} - \frac{{\Delta \omega }}{2}} \right) = {A^2}({{\omega_0}} )+ \frac{1}{4}[{A({{\omega_0}} )A^{\prime\prime}({{\omega_0}} )- {{({A^{\prime}({{\omega_0}} )} )}^2}} ]\Delta {\omega ^2} + o({\Delta {\omega^2}} )$$

Through comparing Eq. (13) and Eq. (10), the measurement uncertainty induced by the approximation in Eq. (8) can be calculated as

(14)$$\Delta A({{\omega_0}} )= \frac{{\Delta \omega }}{2}{[{A({{\omega_0}} )A^{\prime\prime}({{\omega_0}} )- {{({A^{\prime}({{\omega_0}} )} )}^2}} ]^{\frac{1}{2}}}$$

where the infinitesimal high-order expansion error of $o({\Delta {\omega^2}} )$ is ignored. Similarly, through using Taylor expansion, the measurement uncertainty of the group delay response can be calculated as

(15)$$\Delta GD({{\omega_0}} )= \frac{{GD^{\prime\prime}({{\omega_0}} )}}{{24}}\Delta {\omega ^2}$$

where $GD^{\prime\prime}(x )$ represents the 2^nd-order differential of $GD(x )$. According to Eqs. (14) and (15), it can be concluded that the measurement uncertainty of the magnitude and the group delay is determined by the passband width of the ODUT and the fixed frequency interval of the TTOS. For a narrow-band ODUT, the 1^st- and 2^nd-order differentials of either magnitude or group delay responses are large. Therefore, $\Delta \omega $ should be properly reduced to decrease the measurement uncertainty.

An experiment is carried out to verify the above analysis, where the FFP-I is used as the ODUT. Figure 6(a) and (b) present the magnitude and group delay response measurement results by using TTOSs with different frequency intervals, i.e., 6 MHz, 12 MHz, 48 MHz and 96 MHz. It can be seen from Fig. 6 that a small frequency interval of the TTOS is beneficial for improving the frequency response measurement accuracy.

Fig. 6. Measured (a) magnitude (b) group delay responses of the FFP-I by using TTOS with different frequency interval.

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4.2 Measurement uncertainty introduced by the EVNA

As the frequency interval of the TTOS decreases, the measured frequency also decreases. In such a case, the phase measurement error of the EVNA $\Delta \varphi $ puts a limit to the minimum frequency interval of the TTOS, which will be transferred to the measurement uncertainty of the group delay as

(16)$$\Delta GD = \frac{{\Delta \varphi }}{{\Delta \omega }}$$

The datasheet of the employed EVNA indicates that the minimum $\Delta \varphi $ is about 0.3 degree. In the experiment of measuring the FFP-TF, the minimum $\Delta \varphi $ will lead to a group delay uncertainty about 0.008 ns, i.e., 0.3°/360°/100MHz. This measurement uncertainty contributes to the small ripples in the measured group delay response as shown in the inset of Fig. 3(b). Equation (16) indicates that this kind of group delay response measurement uncertainty can be reduced by increasing the frequency interval of the TTOS. This is verified by the inset of Fig. 6(b), where the measured group delay response in the passband is smoother by using a TTOS with a larger frequency interval. In addition, $\Delta \varphi $ increases as the SNR of the received signal decreases. This is the reason why the group delay measurement uncertainty is larger out of the passband.

Based on the above analysis, the measurement procedure can be summarized as follows. Firstly, the frequency interval of the TTOS is set to be as small as possible to obtain the accurate magnitude response of the DUT based on the criterion shown in Fig. 6. Hence, the 3-dB bandwidth the ODUT $\Delta {f_{3dB}}$ can be determined based on the measured magnitude response. Then, based on the measured passband width, a proper frequency interval of the TTOS is set to characterize both the magnitude and the phase response of the DUT with high enough resolution and accuracy. The criterion is based on the following two points. Firstly, the frequency interval of the TTOS should be small enough to finely and accurately characterize the magnitude response of the DUT. In general, the frequency interval of the TTOS should be smaller than ${{\Delta {f_{3dB}}} / {10}}$ to guarantee that the frequency response of the DUT is characterized with a high enough resolution. Secondly, since the directly obtained phase information by using the EVNA is the phase variation within an angular frequency step of $\Delta \omega $, the phase measurement error of the EVNA $\Delta \varphi $ puts a limit to the minimum frequency interval of the frequency interval of the TTOS. The ENVA is generally with a phase measurement error of about 0.3 degree. In addition, the phase variation in the passband of the DUT is generally about 360 degrees. Therefore, the frequency difference of the two-tone radiation should be larger than ${{\Delta {f_{3dB}}} / {100}}$ to guarantee that the phase response can be characterized with a high enough accuracy.

Finally, it should be pointed out that the measurement resolution is ultimately limited by the linewidth of the laser source $\Delta \upsilon $. The linewidth $\Delta \upsilon $ denotes the output frequency instability of the laser source. Therefore, there is no point to set ${{\Delta \omega } / {2\pi }}$ to be smaller than $\Delta \upsilon $ to further enhance the measurement resolution. In addition, the center frequency and the bandwidth of the intermediate-frequency (IF) filter in the EVNA should be set to be equal to ${{\Delta \omega } / {2\pi }}$ and much smaller than ${{\Delta \omega } / {2\pi }}$, respectively. For a small ${{\Delta \omega } / {2\pi }}$, the measurement duration at each point increases. Therefore, the measurement duration increases as the passband width of the ODUT decreases.

5. Conclusion

In summary, a hyperfine OVNA has been proposed and experimentally demonstrated to characterize ultra-narrowband OBPFs. In the OVNA, a TTOS with excellent coherence is generated by applying two single-tone microwave signals with a fixed frequency interval to a DD-MZM. Through finely sweeping the coherent TTOS with a small frequency interval across the passband of the OBPF via electro-optic modulation, the magnitude and phase responses of the OBPF under test can be measured based on fixed low-frequency detection. In the experiment, the magnitude and group delay responses of an FFP-TF and an FFP-I with 3-dB passband widths of 1.5 GHz and 60 MHz, respectively, were successfully measured. Compared with the conventional methods based on SSB and DSB modulation, the proposed OVNA has the following advantages. Firstly, simultaneous two-tone sweeping with a small frequency interval is beneficial for enhancing the SNR of the measurement results in the frequency range of interest, especially for measuring ultra-narrow OBPFs with high out-of-band rejection ratios. Secondly, the heterodyne signal is immune to environmental vibration since the coherent TTOS propagates in a single physical path. Hence, the phase measurement is free of external disturbance. In addition, fixed low-frequency detection is favorable for reducing the cost of the electrical backend. Therefore, the proposed OVNA is a promising candidate to characterize ultra-narrowband OBPFs with high out-of-band rejection ratios, e.g., parity-time symmetric whispering-gallery microcavities [23]. Finally, it should be pointed out that this method is also applicable for characterizing an optical notch filter. However, since the two optical probe tones are located in the notch region in the measurement, the measurement SNR may be greatly deteriorated. For an optical notch filter with a deep notch, the measurement may even be disabled due to the poor SNR.

Funding

National Key Research and Development Program of China (2019YFB2203800); National Natural Science Foundation of China (61421002, 61927821); Fundamental Research Funds for the Central Universities (ZYGX2020ZB012).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Hyperfine optical vector analysis of ultra- narrowband optical bandpass filters based on coherent two-tone sweeping and fixed low-frequency detection

Abstract

1. Introduction

2. Operation principle

3. Experimental results

4. Measurement uncertainty

4.1 Measurement uncertainty induced by the approximation in Eq. (8)

4.2 Measurement uncertainty introduced by the EVNA

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (16)

Optics Express